Question 95
Let $ABC$ be a triangle with $90^\circ\gt\angle ABC\gt\angle BCA$. Let $D$ be the point on the segment $BC$ such that $2\angle DAC=\angle ABC-\angle BCA$. Let $E$ be the intersection other than $A$ of the line $AB$ with the circumcircle of $ACD$, and let $P$ be the intersection of the line $AB$ with the bisector of $\angle BDE$. Similarly, let $F$ be the intersection other than $A$ of the line $AC$ with the circumcircle of $ABD$, and let $Q$ be the intersection of the line $AC$ with the bisector of $\angle CDF$.
Prove that the lines $AB$ and $PQ$ are perpendicular.
Prove that the lines $AB$ and $PQ$ are perpendicular.